How old is that younger rooster? Sounds like he may still be immature. Up to a certain point the more mature will dominate the less mature, but at some point that difference goes away. The dynamics can change as one matures.
Can two roosters share the same coop and not kill each other? It happens. Can two roosters fight until one of them is dead if they share the same coop or run? It happens. You do not get guarantees with behaviors of living animals.
Assuming the younger is mature enough, different things cold happen. They will fight, as you sort of saw. Sometimes that fight is to the death, one just does not stop until the other is dead. Sometimes they fight until one admits defeat, then they work out an accommodation. That accommodation is often that each rooster claims a certain territory during the day out of sight of each other and each gets his own harem. Sometimes they sleep separately, sometimes they share the same coop at night. I have seen two roosters hang together during the day. Different things can happen.
But one thing is as certain as anything can be with chickens, they will know which one is the boss. What often happens when they fight is that one decides he is better running away than staying to fight so he runs away. He admits the other rooster won. There may be some rematches in the future but as long as one runs away it can end peacefully.
This is what concerns me, how much room do you have? Does the loser have enough room to run away? The winner usually chases the loser some. If the loser cannot get away the winner does not know he has won. He keeps attacking. If he can't get away the loser often hunkers down and tries to protect his head. The winner keeps attacking the head because that is the best way to kill him. How much room you have is really important.
It helps if they are raised together, either one a mature rooster and the other a chick or both as siblings, but people do add mature roosters. Sometimes it works even if space is tight, you do not get guarantees with living animals and their behaviors. The more room you have in the coop, on the roosts, and especially outside (as long as they have access to the outside) the better your chances of success. This does not mean you are doomed to failure if space is tight any more than you are guaranteed success with a lot of room. As Mrs. K said, have a Plan B ready and try it when you can observe.
One thing I would not do, do not let them at each other when they are locked in the coop. Give them access to outside so the loser at least has a chance to run away.
Good luck!
